Hi,
I need a good source to understand the derivation for cut-off frequency, relation of input to output by Kirchoff's laws and what is meant exactly by the impulse response of the amplifier (providing it is not made of a simple unity gain buffer).

I found this app note attached but I can't figure out the part highlighted in red.

also in the first part Vi and Vo are considered as grounds in for the ac signal, but considering an ideal op amp, why isn't Z2 and Z3 considered in series for deriving eqn 1.(only z2) (providing op amp have infinite input impedance)

To make a sharp cutoff (instead of a droopy cutoff), a Sallen-Key filter uses some positive feedback to boost the -6dB level at the cutoff frequency of the two RC filters to -3dB. The positive feedback makes the filter ring a little at the cutoff frequency.

I'm assuming this is to get rid of your current source oscillation mentioned in this thread. As I said in post #24, your current source will probably not put out the correct average current when it is oscillating (it didn't in simulation). Did you try putting the series combination of 100nF and 100Ω in parallel with the inductor to kill the oscillation?

So to clarify the situation: We are seeing this (ringing) at the output and for it to occur, the input signal has to contain a harmonic equal to the cutt-off frequency. When we are saying 'start' and 'stop', do we mean when this particular frequency is present at the input and when it is not? .. or is it that the first few cycles pass through until they are stopped by the filter?

Ringing in a Sallen-Key filter does not occur if the filter has a droppy Bessel cutoff. It occurs when the filter has a sharp Butterworth or "Shabbychev" cutoff.
It occurs with any frequency near the cutoff frequency.
It occurs if a tone near the cutoff frequency is suddenly started or stopped.

Audio crossover filters use Sallen-Key filters and the ringing is not heard.
Audiophools are purists who use a Linkwitz-Riley filter that is sort of droopy with no ringing.

The Barkhausen is satisfied because a Sallen-key filter has a low amount of positive feedback. Far lower than is needed to cause continuous oscillation. It is not unstable, it simply has a +3dB peak in its frequency response to boost the -6dB drop at the cutoff frequency caused by the two RC filters.

I found this app note attached but I can't figure out the part highlighted in red.

also in the first part Vi and Vo are considered as grounds in for the ac signal, but considering an ideal op amp, why isn't Z2 and Z3 considered in series for deriving eqn 1.(only z2) (providing op amp have infinite input impedance)

Click to expand...

If you are familiar with Millman's Theorem, you will see that eq. 1 and the portion of eq. 2 that you have outlined in red are both examples of Millman's.